Flatness for pseudo-ASL Gröbner bases
Establish the following flatness assertions: (1) Prove that for any pseudo-ASL (A,≼), the generic algebra of leading terms A_gen is a flat deformation of A; (2) Prove that for any choice of algebra of leading terms (A,≼,A_lt), the algebra A_lt is a flat deformation of A_gen through algebras of leading terms, and A_disc is a flat deformation of A_lt through algebras of leading terms; (3) Prove that if two algebras of leading terms for (A,≼) are each flat deformations of the other, then the induced bijection between their standard monomials is an isomorphism of algebras; (4) Prove that if for two algebras of leading terms A_lt and A'_lt the zero-product sets Z⊆Z', then A'_lt is a flat deformation of A_lt; (5) Prove that for any ideal I⊆A, the quotient A_lt/sm(I) is a flat deformation of A/I.
References
Conjecture (Flatness for pseudo-ASL Gröbner bases). (1) Given $(A,\preceq)$, $A_{gen}$ is a flat deformation of $A$. (2) Given $(A, \preceq, A_{lt})$, $A_{lt}$ is a flat deformation of $A_{gen}$ (through algebras of leading terms), and $A_{disc}$ is a flat deformation of $A_{lt}$ (through algebras of leading terms). (3) Given $(A,\preceq)$, if two algebras of leading terms are each flat deformations of the other, then the natural bijection between their standard monomials is in fact an isomorphism of algebras. (4) Given $(A,\preceq)$ and two algebras of leading terms $A_{lt}, A'{lt}$, let $Z$ (resp., $Z'$) be the set of those pairs of monomials in $\WH \times \WH$ whose products become $0$ in $A{lt}$ (resp., $A'{lt}$). If $Z \subseteq Z'$, then $A'{lt}$ is a flat deformation of $A_{lt}$. (5) Given $(A,\preceq, A_{lt})$ and an ideal $I \subseteq A$, ${A_{lt}{sm(I)}$ is a flat deformation of ${A}{I}$.